Standing waves for a class of Schr"odinger equations with potentials in $L^infty$

We prove the existence of standing waves to the following family of nonlinear Schrödinger equations: [ ih∂_tψ = −h2Δψ + V (x)ψ − ψ|ψ|^{p−2}, (t, x) ∈ R × R^n$ provided that $h > 0$ is small, $2 < p < 2n/(n − 2)$ when $n ≥ 3$, $2 < p < ∞$ when $n = 1, 2$ and $V (x) ∈ L^∞(R^n)$ is assumed to have a sublevel with positive and finite measure